Lyapunov instabilities of Lennard-Jones fluids

Abstract

Recent work on many particle system reveals the existence of regular collective perturbations corresponding to the smallest positive Lyapunov exponents (LEs), called hydrodynamic Lyapunov modes. Until now, however, these modes are only found for hard core systems. Here we report new results on Lyapunov spectra and Lyapunov vectors (LVs) for Lennard-Jones fluids. By considering the Fourier transform of the coordinate fluctuation density $u^{(\alpha )}(x,t)$, it is found that the LVs with $\lambda \approx 0$ are highly dominated by a few components with low wave-numbers. These numerical results provide strong evidence that hydrodynamic Lyapunov modes do exist in soft-potential systems, although the collective Lyapunov modes are more vague than in hard-core systems. In studying the density and temperature dependence of these modes, it is found that, when the value of Lyapunov exponent ${\lambda }^{(\alpha )}$ is plotted as function of the dominant wave number $k_{max}$ of the corresponding LV, all data from simulations with different densities and temperatures collapse onto a single curve. This shows that the dispersion relation ${\lambda }^{(\alpha )}$ vs. $k_{max}$ for hydrodynamical Lyapunov modes appears to be universal irrespective of the particle density and temperature of the system. Despite the wave-like character of the LVs, no step-like structure exists in the Lyapunov spectrum of the systems studied here, in contrast to the hard-core case. Further numerical simulations show that the finite-time LEs fluctuate strongly. We have also investigated localization features of LVs and propose a new length scale to characterize the Hamiltonian spatio-temporal chaotic states.

12 pages, 20 figures

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